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Theorem sbal1yz 1999
Description: Lemma for proving sbal1 2000. Same as sbal1 2000 but with an additional disjoint variable condition on 𝑦, 𝑧. (Contributed by Jim Kingdon, 23-Feb-2018.)
Assertion
Ref Expression
sbal1yz (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
Distinct variable groups:   𝑥,𝑦   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbal1yz
StepHypRef Expression
1 dveeq2or 1814 . . . . . 6 (∀𝑥 𝑥 = 𝑧 ∨ Ⅎ𝑥 𝑦 = 𝑧)
2 equcom 1704 . . . . . . . . 9 (𝑦 = 𝑧𝑧 = 𝑦)
32nfbii 1471 . . . . . . . 8 (Ⅎ𝑥 𝑦 = 𝑧 ↔ Ⅎ𝑥 𝑧 = 𝑦)
4 19.21t 1580 . . . . . . . 8 (Ⅎ𝑥 𝑧 = 𝑦 → (∀𝑥(𝑧 = 𝑦𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
53, 4sylbi 121 . . . . . . 7 (Ⅎ𝑥 𝑦 = 𝑧 → (∀𝑥(𝑧 = 𝑦𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
65orim2i 761 . . . . . 6 ((∀𝑥 𝑥 = 𝑧 ∨ Ⅎ𝑥 𝑦 = 𝑧) → (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥(𝑧 = 𝑦𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑))))
71, 6ax-mp 5 . . . . 5 (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥(𝑧 = 𝑦𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
87ori 723 . . . 4 (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑥(𝑧 = 𝑦𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
98albidv 1822 . . 3 (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑦𝑥(𝑧 = 𝑦𝜑) ↔ ∀𝑦(𝑧 = 𝑦 → ∀𝑥𝜑)))
10 alcom 1476 . . . 4 (∀𝑦𝑥(𝑧 = 𝑦𝜑) ↔ ∀𝑥𝑦(𝑧 = 𝑦𝜑))
11 sb6 1884 . . . . . 6 ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧𝜑))
122imbi1i 238 . . . . . . 7 ((𝑦 = 𝑧𝜑) ↔ (𝑧 = 𝑦𝜑))
1312albii 1468 . . . . . 6 (∀𝑦(𝑦 = 𝑧𝜑) ↔ ∀𝑦(𝑧 = 𝑦𝜑))
1411, 13bitri 184 . . . . 5 ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑧 = 𝑦𝜑))
1514albii 1468 . . . 4 (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥𝑦(𝑧 = 𝑦𝜑))
1610, 15bitr4i 187 . . 3 (∀𝑦𝑥(𝑧 = 𝑦𝜑) ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
17 sb6 1884 . . . 4 ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑))
182imbi1i 238 . . . . 5 ((𝑦 = 𝑧 → ∀𝑥𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑))
1918albii 1468 . . . 4 (∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑) ↔ ∀𝑦(𝑧 = 𝑦 → ∀𝑥𝜑))
2017, 19bitr2i 185 . . 3 (∀𝑦(𝑧 = 𝑦 → ∀𝑥𝜑) ↔ [𝑧 / 𝑦]∀𝑥𝜑)
219, 16, 203bitr3g 222 . 2 (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
2221bicomd 141 1 (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  wo 708  wal 1351  wnf 1458  [wsb 1760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533
This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761
This theorem is referenced by:  sbal1  2000
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